任意阶曲线四边形上的模拟框架
Mimetic framework on curvilinear quadrilaterals of arbitrary order
AI总结：
本文提出基于微分几何和代数拓扑的任意阶模拟离散化方法，通过保持连续与离散操作的可交换性及方向性，实现Hodge分解的完整框架。
摘要：
本文介绍了高阶模拟离散化，这些离散化牢固地植根于变量定义的几何中。论文展示了微分几何中的基本构造在代数拓扑中有离散对应。讨论了在连续微分形式和离散上链之间切换的通用映射，并最终基于投影提出了这些思想的模拟谱元素实现，其中有限维层次上的操作与连续层次上的操作可交换。在连续层次、离散层次上引入了两种定向（内定向和外定向），并证明了新模拟算子保持定向。连续公式与离散代数拓扑设置之间的一一对应，提供了域的有向离散边界的特征。给出了连续、离散和有限维层次上的Hodge分解。它似乎是该框架结构的主要组成部分。
英文摘要：
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in algebraic topology. Generic maps which switch between the continuous differential forms and discrete cochains will be discussed and finally a realization of these ideas in terms of mimetic spectral elements is presented, based on projections for which operations at the finite dimensional level commute with operations at the continuous level. The two types of orientation (inner- and outer-orientation) will be introduced at the continuous level, the discrete level and the preservation of orientation will be demonstrated for the new mimetic operators. The one-to-one correspondence between the continuous formulation and the discrete algebraic topological setting, provides a characterization of the oriented discrete boundary of the domain. The Hodge decomposition at the continuous, discrete and finite dimensional level will be presented. It appears to be a main ingredient of the structure in this framework.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
